Advertisement

Mathematics of Control, Signals, and Systems

, Volume 31, Issue 4, pp 545–587 | Cite as

Geometry and dynamics of the Schur–Cohn stability algorithm for one variable polynomials

  • Baltazar Aguirre-Hernández
  • Martín Eduardo Frías-ArmentaEmail author
  • Jesús Muciño-Raymundo
Original Article
  • 59 Downloads

Abstract

We provided a detailed study of the Schur–Cohn stability algorithm for Schur stable polynomials of one complex variable. Firstly, a real analytic principal \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure in the family of Schur stable polynomials of degree n is constructed. Secondly, we consider holomorphic \(\mathbb {C}\)-actions \(\mathscr {A}\) on the space of polynomials of degree n. For each orbit \(\{ s \cdot P(z) \ \vert \ s \in \mathbb {C}\}\) of \(\mathscr {A}\), we study the dynamical problem of the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\) such that its holomorphic s-time describes the geometric change of the n-root configurations of the orbit \(\{ s \cdot P(z) = 0 \}\). Regarding the above \(\mathbb {C}\)-action coming from the \(\mathbb {C}\times \mathbb {S}^1\)-bundle structure, we prove the existence of a complex rational vector field \(\mathbb {X}(z)\) on \(\mathbb {C}\), which describes the geometric change of the n-root configuration in the unitary disk \(\mathbb {D}\) of a \(\mathbb {C}\)-orbit of Schur stable polynomials. We obtain parallel results in the framework of anti-Schur polynomials, which have all their roots in \(\mathbb {C}\backslash \overline{\mathbb {D}}\), by constructing a principal \(\mathbb {C}^* \times \mathbb {S}^1\)-bundle structure in this family of polynomials. As an application for a cohort population model, a study of the Schur stability and a criterion of the loss of Schur stability are described.

Keywords

Schur stable polynomials Schur–Cohn stability algorithm Principal G-bundles Complex rational vector fields Lie group actions 

Notes

References

  1. 1.
    Aguirre-Hernández B, Cisneros-Molina JL, Frías-Armenta ME (2012) Polynomials in control theory parametrized by their roots. Int J Math Math Sci 2012:1–19.  https://doi.org/10.1155/2012/595076 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aguirre-Hernández B, Frías-Armenta ME, Verduzco F (2009) Smooth trivial vector bundle structure of the space of Hurwitz polynomials. Automatica 45(12):2864–2868.  https://doi.org/10.1016/j.automatica.2009.09.011 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aguirre-Hernández B, Frias-Armenta ME, Verduzco F (2012) On differential structures of polynomial spaces in control theory. J Syst Sci Syst Eng 21(3):372–382.  https://doi.org/10.1007/s11518-012-5197-y CrossRefGoogle Scholar
  4. 4.
    Aguirre-Hernandez B, García-Sosa R, Leyva H, Solis-Duan J, Carrillo FA (2015) Conditions for the Schur stability of segments of polynomials of the same degree. Bol Soc Mat Mex 21:309–321.  https://doi.org/10.1007/s40590-015-0054-x MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alvarez-Parrilla A, Muciño-Raymundo J (2017) Dynamics of singular complex analytic vector fields with essential singularities I. Conform Geom Dyn 21:126–224.  https://doi.org/10.1090/ecgd/306 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ancochea G (1953) Zeros of self-inversive polynomials. Proc Am Math Soc 4:901–902.  https://doi.org/10.1090/S0002-9939-1953-0058748-8 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berenstein CA, Gay R (1991) Complex variables an introduction. Springer, New York.  https://doi.org/10.2307/3618600 CrossRefzbMATHGoogle Scholar
  8. 8.
    Bhattacharyya SP, Chapellat H, Keel LH (1995) Robust control: the parametric approach. Prentice-Hall, Boca Rotan.  https://doi.org/10.1016/S1474-6670 CrossRefzbMATHGoogle Scholar
  9. 9.
    Bonsal FF, Marden M (1952) Zeros of self-inversive polynomials. Proc Am Math Soc 3:471–475.  https://doi.org/10.1090/S0002-9939-1952-0047828-8 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bose NK (1993) Digital filters: theory and applications. Elsevier Sciencie, Nort-Holland.  https://doi.org/10.1137/1030022 CrossRefGoogle Scholar
  11. 11.
    Cohn A (1922) Über die anzahl der wurzeln einer algebrischen glechung in einem kreise. Math Z 14:110–148MathSciNetCrossRefGoogle Scholar
  12. 12.
    Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE (1996) On the Lambert W function. Adv Comput Math 5(4):329–359.  https://doi.org/10.1007/BF02124750 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duistermaat JJ, Kolk JAC (2000) Lie groups. Springer, Berlin.  https://doi.org/10.1017/S0013091501214412 CrossRefzbMATHGoogle Scholar
  14. 14.
    Fam AT, Meditch JS (1978) A canonical parameter space for linear systems design. IEEE Trans Automat Control 23(3):454–458.  https://doi.org/10.1109/TAC.1978.1101744 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gargantini I (1971) The numerical stability of the Schur–Cohn criterion. SIAM J Numer Anal 8(1):24–29.  https://doi.org/10.1137/0708003 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gregor J (1958) Dynamické systémy s regulární pravou stranou I. Pokroky Mat Fyz Astron 3:153–160Google Scholar
  17. 17.
    Gregor J (1958) Dynamické systémy s regulární pravou stranou II. Pokroky Mat Fyz Astron 3:266–270Google Scholar
  18. 18.
    Griffiths P, Harris J (1978) Principles of algebraic geometry. Wiley, New York.  https://doi.org/10.1002/9781118032527 CrossRefzbMATHGoogle Scholar
  19. 19.
    Hansen VL (1980) Coverings defined by Weierstrass polynomials. J Reine Angew Math 314:29–39MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hansen VL (1989) Braids and coverings: selected topics. Cambridge University Press, Cambridge.  https://doi.org/10.1112/blms/23.1.104 CrossRefzbMATHGoogle Scholar
  21. 21.
    Hinrichsen D, Pritchard AJ (2005) Mathematical systems theory I, modelling, state space analysis, stability and robustness. Springer, New York.  https://doi.org/10.1108/03684920510614885 CrossRefzbMATHGoogle Scholar
  22. 22.
    Jury EI (1958) Sampled-data control systems. Wiley, New York.  https://doi.org/10.1036/1097-8542.600300 CrossRefzbMATHGoogle Scholar
  23. 23.
    Katz G (2003) How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties. Expo Math 21(3):219–261.  https://doi.org/10.1016/S0723-0869(03)80002-6 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    López JL, Muciño-Raymundo J (2000) On the problem of deciding whether a holomorphic vector field is complete. In: Complex analysis and related topics (Cuernavaca, 1996), operator theory: advances and applications, vol 114. Birkhäuser, Basel, pp 171–195.  https://doi.org/10.1007/978-3-0348-8698-7_13 CrossRefGoogle Scholar
  25. 25.
    Muciño-Raymundo J, Valero-Valdés C (1995) Bifurcations of meromorphic vector fields on the Riemann sphere. Ergod Theory Dyn Syst 15:1211–1222.  https://doi.org/10.1017/S0143385700009883 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Muciño-Raymundo J (2002) Complex structures adapted to smooth vector fields. Math Ann 322(2):229–265.  https://doi.org/10.1007/s002080100206 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rahman QI, Schmeisser G (2002) Analytic theory of polynomials, London mathematical society monographs 26. Claredon Press, OxfordGoogle Scholar
  28. 28.
    Rudolph L (1983a) Algebraic functions and closed braids. Topology 22(2):191–202.  https://doi.org/10.1016/0040-9383(83)90031-9 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Schur J (1918) Über potenzreihen, die im innern des einheitskreises beschänk sind. Journal für die reine und angewandte Mathematik 1917(147):122–145Google Scholar
  30. 30.
    Strebel K (1984) Quadratic differentials. Springer, Berlin.  https://doi.org/10.1007/978-3-662-02414-0_2 CrossRefzbMATHGoogle Scholar
  31. 31.
    Strikwerda JC (2004) Finite difference schemes and partial differential equations, 2nd edn. SIAM, Philadelphia.  https://doi.org/10.1137/1.9780898717938 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autonoma Metropolitana IztapalapaMexico CityMexico
  2. 2.Departamento de MatemáticasUniversidad de SonoraHermosilloMexico
  3. 3.Centro de Ciencias de MatemáticasUNAMMoreliaMexico

Personalised recommendations